Fourthorder operators on manifolds with a boundary
Abstract
Recent work in the literature has studied fourthorder elliptic operators on manifolds with a boundary. This paper proves that, in the case of the squared Laplace operator, the boundary conditions which require that the eigenfunctions and their normal derivative should vanish at the boundary lead to selfadjointness of the boundaryvalue problem. On studying, for simplicity, the squared Laplace operator in one dimension, on a closed interval of the real line, alternative conditions which also ensure selfadjointness set to zero, at the boundary, the eigenfunctions and their second derivatives, or their first and third derivatives, or their second and third derivatives, or require periodicity, i.e. a linear relation among the values of the eigenfunctions at the ends of the interval. For the first four choices of boundary conditions, the resulting 1loop divergence is evaluated for a real scalar field on the portion of flat Euclidean 4space bounded by a 3sphere, or by two concentric 3spheres.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 April 1999
 DOI:
 10.1088/02649381/16/4/001
 arXiv:
 arXiv:hepth/9803036
 Bibcode:
 1999CQGra..16.1097E
 Keywords:

 High Energy Physics  Theory
 EPrint:
 25 pages, plain Tex. In the revised version, equations (3.11)(3.13) have been amended